Problem: Simplify the following expression: $a = \dfrac{20z^2}{12z^3 - 40z^2}$ You can assume $z \neq 0$.
Answer: Find the greatest common factor of the numerator and denominator. The numerator can be factored: $20z^2 = (2\cdot2\cdot5 \cdot z \cdot z)$ The denominator can be factored: $12z^3 - 40z^2 = (2\cdot2\cdot3 \cdot z \cdot z \cdot z) - (2\cdot2\cdot2\cdot5 \cdot z \cdot z)$ The greatest common factor of all the terms is $4z^2$ Factoring out $4z^2$ gives us: $a = \dfrac{(4z^2)(5)}{(4z^2)(3z - 10)}$ Dividing both the numerator and denominator by $4z^2$ gives: $a = \dfrac{5}{3z - 10}$